Angular Velocity Calculator

Definition and Units

ω is measured in radians per second (rad/s) in SI. Revolutions per minute (RPM or rev/min) and hertz (Hz, for cycles per second) are common alternatives. Conversions: 1 rev = 2π rad, so 1 rev/s = 2π rad/s ≈ 6.283 rad/s. 1 Hz = 1 rev/s = 60 RPM.

\omega = (Δ θ)/(Δ t) \quad [\omega] = \mathrmrad/s

Period and Frequency

For uniform circular motion the time for one full revolution is the period T, and the cycles per second is the frequency f. Both follow directly from \omega.

A record player at 33⅓ RPM turns 33.33/60 = 0.5556 rev/s = 3.490 rad/s, so each LP side takes T = 1/\omega ≈ 1.800 s.

**Conversions:** 1 Hz = 1 rev/s = 60 RPM; 1 RPM = 2π/60 rad/s ≈ 0.1047 rad/s.

T = (2π)/(\omega), \quad f = (\omega)/(2π) = (1)/(T)

Connection to Linear Speed

A point at radius r from the rotation axis moves with tangential velocity v = ω r when the motion is rigid-body rotation. Doubling r at a fixed ω doubles v. This is why a flywheel rim can fly apart at a few thousand RPM — the tangential speed of the rim approaches the elastic limit of the material.

v_\texttan = \omega \, r

Right-Hand Rule

ω is a vector pointing along the rotation axis, direction given by the right-hand rule: curl your fingers in the direction of rotation and your thumb points along +ω. Two rotating objects with opposite ω (like meshing gears) propagate torque in opposite directions.

\vec\omega = (1)/(2) Σ_i \vecr_i × \vecv_i

When ω Changes

If ω changes, the body has angular acceleration α = Δω / Δt. Under constant α you get rotational kinematics that mirror linear ones: ω = ω₀ + α t and θ = θ₀ + ω₀ t + ½ α t². Variable-torque motors (during spin-up) ride on this throughout.

α = (Δ\omega)/(Δ t)

Converting Between RPM, Hz and rad/s

Three units describe the same rotation and mixing them is the most common source of error. Revolutions per minute is what machinery nameplates use; hertz is revolutions per second; radians per second is what every physics formula requires.

To convert, divide RPM by 60 to get Hz, then multiply by 2π to get rad/s — so rad/s = RPM × π/30, a factor of about 0.1047. A 3000 RPM motor turns at 50 Hz and 314 rad/s. Going the other way, 1 rad/s is about 9.55 RPM. Always convert before substituting into v = \omega r, a_c = \omega^2 r or L = I\omega: those formulas are only valid in radians, because the radian is defined so that arc length equals angle times radius.

\omega\ [\mathrmrad/s] = \frac2π\, N_\mathrmRPM60 = 2π f

Centripetal Acceleration and Real Limits

Any point on a rotating body accelerates towards the axis at a_c = \omega^2 r, even at constant angular speed, because its direction is continually changing. The square on \omega is why rotational speed limits are so unforgiving: doubling the RPM quadruples the stress on the rim.

The consequences are practical. A grinding wheel rated for 6000 RPM is not merely inefficient at 12000 RPM — it is four times closer to bursting. A washing machine drum at 1400 RPM subjects the load to around 400 g at a 0.18 m radius, which is what drives the water out. And a centrifuge is specified in "relative centrifugal field", a multiple of g, precisely because the acceleration rather than the speed is what separates the sample.

a_c = \omega^2 r = (v^2)/(r)

Angular Momentum and Conservation

A rotating body carries angular momentum L = I\omega, and with no external torque that product stays constant. Because the moment of inertia I depends on how mass is distributed, changing your shape changes your spin rate.

This is the familiar figure-skater effect: pulling the arms in reduces I, so \omega must rise to keep L fixed. The same conservation law explains why a collapsing gas cloud spins up into a fast-rotating star, why a cat can right itself in mid-air without any external torque, and why a spinning bicycle wheel resists being tilted. Note that kinetic energy is *not* conserved in the skater example — the skater does work pulling the arms inward against the centripetal requirement, and that work shows up as extra rotational energy.

L = I\omega = \textconstant, \qquad K_\textrot = \tfrac12I\omega^2

Worked Example

Inputs

• angle: 360
• time: 2

Result: ω = 3.1416 rad/s (180 °/s); period T = 2 s; frequency f = 0.5 Hz

Explanation

A disc that completes one full revolution in 2 seconds sweeps 360° (= 2π rad) in Δt = 2 s, so ω = 2π/2 = π ≈ 3.1416 rad/s. Converting: 360° / 2 s = 180°/s. Because the rotation repeats every 2 s, the period is 2 s and the frequency is 0.5 Hz (or 30 RPM). This pattern — one turn per 2 s — is exactly what you get from a standard 30 RPM turntable at 33⅓ RPM turned down via a reduction gear.

Second Scenario

Inputs

• angle: 432
• time: 2

Result: ω = 3.1416 rad/s (180 °/s); period T = 2 s; frequency f = 0.5 Hz

Explanation

This scenario uses different inputs (angle = 432, time = 2) to show how changing one variable affects the angular velocity result. Run the calculator above with these values to get the exact updated output with step-by-step work.